[ayerjkcn] discriminating paraboloids

z = a*x^2 + b*x*y + c*y^2

compute the discriminant d = b^2 - 4*a*c .  (why is the discriminant the same as that of the quadratic equation?)  (normally, this discriminant is used to distinguish between 2D conic sections, i.e., between ellipse, parabola, and hyperbola.  here we are using it to distinguish between 3D quadric surfaces.  why does the same discriminant still work?)

if d is negative, the surface is an elliptic paraboloid, the easier to understand shape.  a and c overpower the weird cross term.  assuming a and c are positive, then all points except the origin are positive.  the origin is zero.

if d is positive, the surface is a hyperbolic paraboloid.  the weird cross term dominates.  in some directions function goes up; in others, down.

if d is zero, then it is a parabolic cylinder, a parabolic trough.  the bottom of the trough is tangent to the xy plane along a line through the origin.  included in this case is the plane z=0, but that is easy to recognize: a=b=c=0.

to do: prove all this by rotating the axes to eliminate the xy term (cross term).  2*theta = atan(b/(a-c)).  better done with atan2 to deal with a=c.  sin(2*theta) and cosine can be gotten by setting up a right triangle, then theta by the half-angle formulae.